Examples for Problem Solving with Quadratic Equations

“ Solving Problems with Quadratic Equations” KEY

(1) Kelsey jumps from a diving board, springing up into the air and then dropping feet first toward the pool. The distance from her toes to the water’s surface in feet t seconds after she leaves the board is d = -16t2 + 18t + 10.

(a) What is the maximum height of Kelsey’s toes during this jump? When does the maximum height occur? 15.06 feet, 0.56 seconds (b) When do Kelsey’s toes hit the water? 1.53 seconds (c) What might the constant term 10 in the equation tell you about Kelsey’s jump? Kelsey jumps from a board 10 feet above the water. The constant represents that initial height.

(2) A square has sides of length x centimeters. A new rectangle is created by increasing one dimension of the square by 2 centimeters and increasing the other dimension by 3 centimeters. (a) Make a sketch and label the area of each of the four sections created by increasing the dimensions.

2 2x 6

x x2 3x

(b) Write two expressions, one in factored form and one in expanded form, for the area of the new rectangle. (x + 2)(x + 3) x2 + 5x + 6

(c) Use your expressions from part (b) to write two equations for area, A, of the rectangle. Graph both equations on your calculator. Compare these graphs with the ones you made in the problem before. A = (x + 2)(x + 3) A = x2 + 5x + 6

Although the graph is shifted to another position in the plane, the two equations still produce the same parabola. (3) The dot patterns below represent the first four triangular numbers.

1st 2nd 3rd 4th

(a) Look for a pattern. Use the pattern to help you make a table of the first ten triangular numbers.

Term Number Number of Dots 1 1 2 3 3 6 4 10 5 15 6 21 7 28 8 36 9 45 10 55

(b) Write an equation for calculating the nth triangular number. t = n(n + 1) 2 (c) Use your equation to predict the 15th triangular number. 120 (d) Does your equation represent a quadratic relationship? Explain. It does represent a quadratic relationship because when you multiply out the expression for finding the nth triangular number, you see that its highest power term is n2.

(4) A large cube with edges of length n centimeters is built from centimeter cubes. The faces of the large cube are painted.

(a) Write an equation for the number of centimeter cubes in the large cube. c = n3 (b) Write an equation for the number of centimeter cubes painted on: (i) three faces (ii) two faces (iii) one face (iv) no faces c = 8 c = 12(n – 2) c = 6(n – 2)2 c = (n – 2)3